Browse Questions

# Show that the function = 2x - | x | is continuous at x = 0.

Toolbox:
• If $f$ is a real function on a subset of the real numbers and $c$ a point in the domain of $f$, then $f$ is continous at $c$ if $\lim\limits_{x\to c} f(x) = f(c)$.
• Every polynomial function $f(x)$ is continous.
Step 1:
Let $f(x)=2x-\mid x\mid$
We have $f(x)=2x-\mid x\mid$
$\Rightarrow f(x)=\left\{\begin{array} {1 1}2x-(-x)&if\;x<0\\2x-(x)&if\;x>0\end{array}\right.$
Continuity at $x=0$
We have LHL as
$\lim\limits_{x\to 0^-}f(x)=\lim\limits_{x\to 0^-}2x+x$
$\Rightarrow 0$
Step 2:
We have RHL as
$\lim\limits_{x\to 0^+}f(x)=\lim\limits_{x\to 0^+}2x-x$
$\Rightarrow 0$
Since the LHL =RHL,the given function $2x-\mid x\mid$ is continuous at $x=0$